Lessons of the École Doctorale de l'Institut de Physique du Globe de Paris
and of the École Doctorale de l'Observatoire de Paris

Follow this link to have information on the two Écoles Doctorales

Seminar (post-master course):

Gravimetry, Relativity, and the Global Navigation

Satellite Systems (Galileo and GPS)

Albert Tarantola (web page, e-mail) & Bartolomé Coll (web page, e-mail)

with the contribution of José-Maria Pozo (e-mail)

Photographs 2004-2005

News:
• Click here to download the program of the seminar.
• Please click here to see the list of students.
• Some texts of possible interest are posted here.

Lecture Notes:
• Introduction to Tensor Calculus (slides).
• Flash introduction to Special Relativity (slides).
• The first two lessons (text).
• Third lesson Introduction to Differential Geometry (text) (some slides).
• An excellent small text on Differential Geometry, written by our dear friend André Heslot (1954-2000): Géométrie Différentielle.
• Flash introduction to Gravimetry (text).
• Relativistic Corrections in the GPS System (slides).
• Introduction to General Relativity I (slides).
• Introduction to General Relativity II (Schwarzscild, Kerr, UAI coordinates) (slides).
• UAI Resolution on Reference Frames (English) (French).
• Coordinate Systems in (General) Relativity (slides).
• Emission Coordinates as Positioning Systems (slides).
• Flash introduction to Inverse Problems (slides). Note that Tarantola's book on Inverse Problems is available for free download (tarantola.cc).
• Formulation of the Relativistic Positioning and Gravimetry Problem (slides) (text).

Rationale:

While today's presentations of gravimetry still use a Newtonian veiwpoint of the space-time, the gravity field is, and only is, the space-time metric. Gravimetry is gravitation, and gravitation theory is the science of the geometry of space-time. In this course, we shall examine the conceptual foundations of relativistic gravimetry. Which kind of (relativistic) coordinates are accessible to experiment? Which kind of measurements may bring information on the space-time metric? The development of the theory has strong implications for the Global Navigation Satellite Systems (Galileo or the GPS): while these systems are today operated as if the space-time was Newtonian, using relativity just as a set of "corrections", one may imagine an entirely, fully relativistic, way of operation. Such a system would constitute the ultimate gravimeter. The emphasis of the course will be in the theory, not in the applications.

Requirements for the course:

The introductory part may have a very variable length, to adapt to the median background of the students, but some familiarity with special relativity and tensor calculus will be of great help. This will essentially be a theoretical seminar. Although we shall have a very precise application in mind (the GNSS and gravimetry) the theory to be developed shall not have an immediate use: present day practices are quite far from being based in good theory. We do think that, some day, the GNSS systems and the gravimetric satellites will be operated in this way, but it may take a long time before the proposed ideas come into normal use.

Only theoretically minded students should attend the course. If you don't like theory or you don't like mathematics, please don't try to attend this seminar.

Approximately 20 hours.

This is a link to the other teachings of A.T. at the Institut de Physique du Globe de Paris